Laminar fluid flow in concentric annular ducts of non-conventional cross-section applying GBI method

Fluid flow in concentric or eccentric annular ducts have been studied for decades due to large application in medical sciences and engineering areas. This paper aims to study fully developed fluid flow in straight ducts of concentric annular geometries (circular with circular core, elliptical with circular core, elliptical with elliptical core, and circular with elliptical core) using the Galerkin-based Integral method (GBI method). The choice of method was due to the fact that in the literature it is not applied in ducts of cross-sections of the annular shape with variations between circular and elliptical. Results of different hydrodynamics parameters such as velocity distribution, Hagenbach factor, Poiseuille number, and hydrodynamic entrance length, are presented and analyzed. In different cases, the predicted hydrodynamic parameters are compared with results reported in the literature and a good concordance was obtained.

Several solution techniques have been developed, over the years, to obtain information about hydrodynamic parameters that characterize the flow in ducts with different cross-section geometries. Alegria et al. (2012), presented an analytical and numerical study for the isothermal flow of Herschel-Bulkley viscoplastic fluid using three duct configurations of elliptical cross-sections: elliptical duct, concentric annular elliptical and circular eccentric. This study aimed to analyze the effect of geometric and rheological parameters of the fluid with pressure drop. Alves et al. (2014), presented a hybrid analytical-numerical solution for the isothermal hydrodynamic problem of the fully developed laminar flow in circular concentric annular ducts using the Generalized Integral Transform Technique (GITT).
In this work, the authors applied a conform transform, together with the GITT and presented results for Poiseuille number, Hagenbach factor, and velocity profiles.
The Galerkin-based integral (GBI) method has well-known applicability in flow analysis in straight ducts with arbitrary geometry (Lee & Kuo, 1998;Lee & Lee, 2001;dos Santos Júnior et al., 2020;Santos Júnior, 2018;Haji-Sheikh & Beck, 1990;Lakshminarayanan & Haji-Sheik, 1992). However, it can also be used in other engineering areas (Franco et al., 2016;Pessôa et al., 2017;Santos et al., 2015;Santos et al., 2012;Lima et al., 2004;Franco et al., 2019;Lima et al., 2014). Dos Santos Júnior et al. (2020), present a theoretical study applied to fully developed internal laminar flow through a corrugated cylindrical duct, using the Galerkin-based integral method. As an application, the authors have used the proposed formulation to predict heavy oil flow at different operating conditions considering temperature-dependent viscosities ranging from 1715 to 13000 cP. In this research are reported many fluid dynamics parameters, such as the Fanning friction factor, Reynolds number, shear stress, and pressure gradient. Research, Society andDevelopment, v. 10, n. 1, e10710111547, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i1.11547 3 Given the real importance, the range of work, the applicability of annular ducts and also the shortage of results in the analysis of fluid flow in ducts with annular elliptical geometry, this work aims to study the fluid dynamic behavior of incompressible and Newtonian fluid in non-conventional geometries using the Galerkin-based Integral Method. Four geometries were used: circular with circular core, elliptical with circular core, elliptical with elliptical core, and circular with elliptical core.
Although the fluid flow theory is well understood very few studies have attempted to predict fluid flow in duct with annular geometry using the GBI method. Thus, it is clear, the innovative contribution of this research: fluid flow in nonconventional and annular geometries not yet fully understood and several physical situations remain unsolved. In essence, this is purely quantitative research, according to Pereira et al. (2018).

Methodology
For an appropriated mathematical formulation related to fluid flow in ducts, the following assumptions are given:

The Geometry and base functions
For mathematical analysis we choose the cylindrical coordinate system ( , , ). The concentric annular geometries to be analyzed will be: circular with core circular and elliptical with core elliptical. For these geometries we will use the following parameterizations: 1 ( ) = ±√ 1 2 1 2 1 2 cos 2 ( ) + 1 2 sin 2 ( ) and 2 ( ) = ±√ 2 2 2 2 2 2 cos 2 ( ) + 2 2 sin 2 ( ) where and , with = 1,2, are the axes of the ellipse, when ≠ and the radius of the circumference when = , with 0 ≤ ≤ 2 , as illustrated in Figure 1. This way, it is possible to obtain large varieties of geometries for the duct combining only these parameters.  (2020) Thus, considering the cross-section given by the curves 1 ( ) and 2 ( ), the cross-section area of the duct is given by:
Considering the parameterization in polar coordinates, 1 ( ) and 2 ( ), we define the following functions: Thus, the set of bases functions will be as follows: Putting Equation (5) into equation (6), we can write base functions as follows: ( , ) = ( , ) with , = 0, 1,2, … , where the subscript represents the -th term of the set of base functions in Equation (6).

Linear momentum equation and solution methodology
Now, considering the hypotheses cited before, the linear momentum equation written in the cylindrical coordinate system, is given as follows: with = 0 in Γ. In Equation (9) were considered the following dimensionless parameters: where is the characteristic length.
Since the mean velocity of the fluid is given by: the normalized dimensionless mean velocity can be written as follows: where is the dimensionless mean velocity of the fluid.
In this research, the solution of the momentum equation was obtained by using Galerkin-based integral method (GBI method). This method isn´t new, but never have been applied to predict fluid flow in ducts with non-conventional cross-section ducts.
By the GBI method, the solution is approximated as a linear combination of a set of base functions. Thus, for the Equation (9), we can written: The functions 1 , 2 , … , are linearly independent functions and satisfy the same homogeneous boundary conditions of .
Applying the GBI method in Equation (9) and replacing W as in the Equation (14), we obtain the following matrix system: Since that we obtain the coefficients 1 , 2 , . . . , of Equation (14).
The Reynolds number is defined as: A parameter of great interest in fluid mechanics is the Poiseuille number. It is defined as the product of the Reynolds number and the friction factor, as follows: with, ℎ = ℎ is the dimensionless hydraulic diameter.
The number of increments in the pressure drop or Hagenbach factor, K (∞), according to reference Shah & London (1978), is defined as: or, where (∞) and (∞), are respectively the kinetic energy correction factor and the momentum flux correction factor, given by: and The dimensionless hydro-dynamic entry length, ℎ + , according to Shah & London (1978), is given by: where = is the maximum normalized dimensionless velocity, is the maximum dimensionless velocity.

Numerical Computation
For obtain the numerical results, a computational code in the software Maple Student Edition was developed. Following, the flowchart shows the calculation procedure using the GBI method: Source: Authors (2020).

Results and Discussion
In this section, we present the numerical results obtained for different hydrodynamic parameters (Poiseuille number, Hagenbach factor and dimensionless entry length) obtained for different aspects ratio of the geometries presented in Figure 2. In the simulations, we consider the characteristic length = 1 corresponding to the radius of the outer circumference or half of the major axis of the outer ellipse.

G1 geometry
For G1 geometry, (Figure 2a), consider the aspect ratio given by = 2 1 , with 1 = 1 = 1 and 2 = 2 , as illustrated in Figure 4a. Figure 4b illustrates the velocity field of the fluid inside the duct over the entire cross-section. It can be verified that velocity is zero at the wall and increasing towards the central region of the annular. This behavior occurs because of the noslip condition established in the simulations. The friction effect is transmitted for all layers of fluid that are adjacent (formation of the hydrodynamic boundary layer).
In Table 1, we present the values of the hydrodynamics parameters obtained of the simulations. We take the aspect ratio = 2 1 ranging from 0.01 to 0.9. From the analysis of this table, we can see that increased values of the aspect ratio β, the Poiseuille number has increased. In contrast, all others parameters have been reduced. When → 0, we have the parameterization and the value of the Poiseuille number, = 16, of the circular duct. On the other hand, as the > 1, the system becomes two parallel flat plates with almost null thickness. Furthermore, in Table 1 velocities peaks are observed clearly. To assure the accuracy of the values presented in Table 1, the number of base functions ranged from = 5 to = 15.
When the aspect ratio tends to be = 1, more base functions are needed to better approximates the values predicted by simulations with that reported in the literature. The proposed model and the solution procedure were validated with previous works reported in the literature. Figure 5-7 show these results. From the analysis of there figures, we can see that on excellent agreement was obtained, except for small number of the aspect ratio, where some discrepancy are seen.   Source: Authors (2020).
It was observed a large computational effort in obtaining the hydrodynamic parameters when the aspect ratio is close to 0 and 1.

G2 Geometry
For G2 geometry, Figure 2(b), two aspect ratios are considered: the first concerns to 1 ( ), with 1 = 1 1 , 1 = 1, and 1 = 0.9, 0.7 and 0.5; the second concerns to the relationship between the curve 2 ( ) and 1 ( ), that is, = 2 1 and 2 = 2 as the curve 2 ( ) is a circumference. Also, it is notable that the fluid velocity is very sensitive to the shape of the duct. A peak of maximum velocity is clearly observed, and its magnitude is different for each geometry. The peak location in the cross-section of the duct varies accordingly with the established geometry. For G2 geometry, maximum and minimum velocities of 2.7116 and 0.2456, respectively, were obtained.
In Table 2, we present the values of the parameters obtained in the simulations.  Table 2, there were variations in the number of chosen base functions ranging from = 3 to = 15. We state that results and discussions with the same geometry are presented in previous works , when the behavior of the normalized dimensionless velocity profile is observed geometrically using other methodologies. Figure 9-13 illustrate the behavior of the hydrodynamic parameter as a function of the parameter. Figures 9 and 10 show the results reported in Table 2, and some numerical values for found in the literature are used for comparison purpose.
Analyzing the figures, we can see that a good agreement was except for the smaller parameter , which can be attributed to the fact that a larger number of base functions, consequently a longer computational time, are necessary for the convergence of the values. This increase in base functions can significantly difficult the calculation of the inverse matrix −1 that makes up the matrixial system in Equation (18).  Source: Authors (2020). Research, Society andDevelopment, v. 10, n. 1, e10710111547, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i1.11547  Source: Authors (2020).
In Figures 12 and 13, we present the graphs presented in Table 2 for the values of (∞) and ℎ + . From the analysis of these figures we can observe an increase in the parameters with increased parameter (fixed 1 values) and decreasing with increased 1 parameter (fixed values).  Research, Society and Development, v. 10, n. 1, e10710111547, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i1.11547

G3 Geometry
For the G3 geometry (Figure 2c), here we considered confocal ellipses, with two aspect ratios: the first concerns to 1 ( ), with 1 = 1 1 , 1 = 1, and the second concerns to relationship between the curves 2 ( ) and 1 ( ), that is, = 2 1 . Since that the ellipses are confocal, we have the following relationship 1 2 − 1 2 = 2 2 − 2 2 . Figure 14 shows the velocity profile of the fluid in a cross-section area of the duct. From the analysis of this figure, we can see an almost similar behavior find to G2 geometry, however for this case smaller variations in the velocity with the angular coordinate are seen. Source: Authors (2020).
In Table 3, we present the values of the hydrodynamic parameters obtained in the simulations. By analyzing the results reported in Table 3, one can verify that the values of Poiseuille number, and ℎ + increase with the increased parameter, and meanwhile and (∞), parameter have decreased as increased. This behavior occurs for any fixed value of the 1 aspect ratio. However, for fixed value of the parameter, Poiseuille number and have increased as 1 increased. For the values presented in Table 3,    Source: Authors (2020).
In Figures 17 and 18, we present the parameters (∞) and ℎ + as a function of the aspect ratio for two values of the 1 parameters.

G4 Geometry
Now, we consider the G4 geometry, that corresponds to circumference with concentric elliptical core, Figure 2(d). The following aspect ratios were considered: the first concerns the relationship between the major axis of 2 ( ), with the radius of the circumference, that is, 1 = 2 1 , and the second concerns the relationship between the minor axis of 2 ( ), with the radius of the circumference, = 2 1 . For convenience, here we consider 1 = 1 = 1. Figure 19 shows the dimensionless velocity profile inside the duct, for one specified geometrical condition. Similarly, to other cases, a peak of velocity is verified in the annular region. This behavior is in accordance with the reported data in the literature (Lima et al., 2004). Source: Authors (2020).
In Table 4, we present the values of the hydrodynamic parameters obtained in this research. From the analysis of this  For the values presented in Table 4, we consider nine base functions ( = 9).    Figure 20, we can verify that an increase in the parameter provokes an increase in the Poiseuille number, for any fixed 1 parameter. On the other hand, for the fixed value of the parameter, a reduction in the 1 parameter provoked an increase in the hydrodynamic parameter. Furthermore, for lower 1 value, the Poiseuille number has an almost linear behavior, and for higher 1 value this parameter presents a minimum value for a specific value.
By analyzing Figure 21, we can state that the behavior of the Hagenbach factor is in contrast with the Poiseuille number, increasing as the 1 parameter is increased, for a fixed value. Similar behavior was observed for the dimensionless hydrodynamic entry length.
Based on the results reported in Table 4 and Figure 20 we state that they are in accordance with the results reported in Table 1 (G1 geometry), as → 1, that is equivalent to the case as → 1 and, thus, validating all procedures used in this research.

Conclusions
The Galerkin-based integral method in a two-dimensional approach is used for investigating the laminar flow fluid in doubly-connected ducts (circular with circular core, elliptical with circular core, elliptical with elliptical core, and circular with elliptical core). It is demonstrated that the velocity profile, Hagenbach factor, Poiseuille number, and fluid dynamic entry length, can be effectively predicted by this technique. The results, when compared with the literature data, showed good concordance, thus giving a robust applicability of the method and the parameterizations considered for the geometries studied here.
Depending on the duct geometry, aspect ratio, and the number of base functions considered, we have variations in computational effort and accuracy in the obtained results. The study showed that, especially for elliptical core geometries, to use a smaller number of base functions is more advantageous. For smaller aspect ratio, for example, 0 < < 0.1 and even 0 < < 0.4, higher computational time and reduced accuracy were verified. Further, the obtained results show that the dimensionless velocity distribution presents a peak in the annular region which location is dependent on the duct shape (geometry and aspect ratio).
Finally, the GBI method may be used to predict fluid flow behavior in a duct of different shapes, in which solutions or available data do not exist.
In view of the applicability of the Galerkin-based integral method in unconventional geometries, the authors recommend to study fluid flow in duct of different geometries, in which solutions or available data do not exist, and in circular ducts with incrustations, which are originated of paraffin depositions in oil transportation, or solid particles in areas of basic sanitation.